Some polynomials over $\mathbb Q(t)$ and their Galois groups

Examples of polynomials with Galois group over $\mathbb Q(t)$ corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over $\mathbb Q$ through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.